Can someone explain this one (both a and b... I can't get the algebra to work out)?
I don't know how to explain that they are counting the same thing, but I can explain the algebra. (update): I've been thinking, and perhaps soneone else can follow my line of thinking and confirm: imagine you have n pairs of socks [for example purpose lets say 4 (red, blue, green, and yellow)] all with an left and right. Now the right side is saying "how many ways can I pick two socks (a pair) from the 2n (8) total?" Now the right side is a little more tricky because it looks at how to pick pairs. The n^2 comes from how many ways can I choose a pair containing both a left sock and right sock or n*n (4*4) plus I have to consider the pairs of two left socks and the pairs of two right socks. So that would be n*n + C(n,2) + C(n,2) or "simplified" to 2*C(n,2)+n^2. Again double checking the math with the example of 4 pairs, the left side would be C(8,2) = (8*7)/2 = 28. The right side would be 2*C(4,2)+4^2 = 2*4*3/2 + 16 = 12+16= 28 (CHECK)
First, looking at the left side you have C(2n,2) which equals $ \frac{(2n)!}{(2!*(2n-2)!} $. This can be "simplified" to $ \frac{(2n*(2n-1))}{2} $. which further simplifies down to $ 2n^2-n $. The right side starts at $ 2*C(n,2)+n^2 $. so $ 2*(\frac{n!}{(2!*(n-2)!)})+n^2 $. Simplifing this leads to $ \frac{(2*n*(n-1))}{2} + n^2 $ which goes to $ n^2 - n + n^2 $ which ofcourse equals $ 2n^2 - n $. (someone that understands how to make the formulas look nice, feel free to do so).
